When Kan extensions don't exist Kan extension are an incredibly useful concept when they exist. My question is: can we still derive information about a functor when Kan extensions don't exist, and if so, what information and in what way?
I can think of one way to look at it:
Let $p:C\to C'$ and $F:C\to D$ be functors.  We can define a category of pairs $(G:C'\to D, \eta: F\Rightarrow p^*G)$ with morphisms being natural transformations that commute with the $\eta$s.
A left Kan extension of $F$ along $p$ is an initial object in this category, so one can study this category as a replacement when the Kan extension fail to exist.
Has this direction been explored before?
 A: Yes, directions like this have been explored, for all kinds of objects with universal properties (which includes Kan extensions, since as MacLane famously wrote "all concepts are Kan extensions").  Rather than the category whose initial object would be the missing object, it's more common to consider the functor for which a representation would be the missing object.  Properties of this functor can then be considered as properties of the "missing object", and at the extreme the category of all functors/presheaves (or a subcategory such as a category of sheaves, concrete sheaves, schemes, etc.) can be studied as a "better-behaved" replacement for the original category in which "missing objects" have been put in.
As one concrete example, a non-existing left adjoint can exist as a pro-adjoint; in fact it does so precisely when the putative right adjoint preserves finite limits (whereas, modulo size considerations, for an actual left adjoint to exist, the putative right adjoint would have to preserve all limits).
A: There are several points worth making here.
(i) One solution is to use Benabou's theory of 'distributeurs' also called profunctors. A functor F from C to D defines two profunctors basically $D(F-,-):C^{op}\times D\to Sets$ and $D(-,F-): D^{op}\times C\to Sets$. The profunctors from C to D form a bicategory. Any functor considered as a profunctor has an adjoint profunctor (I leave it up to the reader to work this out.) The n-Lab page on profunctors gives several suitable references, and a basic introduction. The Kan extensions question is then solvable as the 'existence' is a question of representability. i.e., you get a profunctor as the Kan extension but do not know if it has a representing object /  functor etc. (Some of that is described in my book (with Jean-Marc Cordier) on Shape Theory: Categorical Methods of Approximation, (Published by Dover). PS. I do not get any royalties from the sales of that book so this is not really self promotion!)
This theory also works in an enriched category setting, and in a derived form (suitably adapted).
(ii) It is worth pointing out that in many contexts you can study the properties of a Kan extension object even if the object does not exist in the category you would like it to be in. This just extends the use by Grothendieck etc. of functors that encode something useful but may not be representable. (You may have some setting in which, say, a moduli space does not exist but there is a functor that behaves as if it was maps from that non-existent object to whatever. You may be able to determine the cohomology of the non-existent moduli space! Alternatively you can enlarge the category of `spaces' so as to think of the functor as being representable but in that larger category. This led to Artin's algebraic spaces and further on to the study of stacks, etc.)
(iii) Another instance of the same sort of thing is with the use pro-objects. I will illustrate this with adjoints rather than with  more general Kan extensions. The inclusion of, for instance, the category of finite groups into that of all groups does not have a left adjoint, but it does have a pro-adjoint which send a group to its filtering system of finite quotients.  Using this sort of idea one can quite often get a pro-version of Kan extensions. (There are several instances of this in quite classical algebraic topology and algebra; again my book with Cordier looks at some of that. I only mention it because I know its contents, and not because those examples are the only important ones. There are many other uses for these ideas.)
For the Kan extensions the idea is that you go via the comma category approach, giving a functor to one of the categories concerned,  and of which you would normally take a limit or colimit. However the limit might not exist.  If the comma category is `filtering' then the resulting functor gives a pro-object that replaces the hoped for Kan extension by an inverse system of objects that give approximations to the Kan extension.
(Above I have used adjoints as examples rather than Kan extensions in the above, as they provide easier instances of the main ideas.)
